Anybody from anyplace can "become" an American. But to be French (or any other European nationality) involves race and ancestry. The Germans realize this, but the French actually go the extent of pretending that anyone born in their country is descended from the ancient Gauls. Ah, yes, but the French are composed of both the Gauls and the Franks.
"..the French are divided into Franks and Gauls. Franks, serious and rather cold, the builders and the Gauls, adorably frivilous, the destroyers of this nation." --Nancy Mitford," The Sun King"
You are correct, I forgot about the Frankish segment of the population. The French don't really emphasize it that much, maybe because the Gauls were the inhabitiants before Roman conquest and made famous by Caesar's "Gaulic Wars", and I think the Franks were later barbarian invaders (Huns and Goth types).
But somehow, when reading your reply all I could first think of was Franks and Beans. Not haute cuisine, but it will have to do.